Question

what is the equation of the graph below? you can click the picture to zoom in!
$y = -|x - 2| + 4$
$y = |x - 2| + 4$
$y = |x - 2| - 4$
$y = -|x + 2| + 4$

Explanation:

Step1: Identify vertex form of absolute value

The general form of an absolute value function is \( y = a|x - h| + k \), where \((h, k)\) is the vertex. From the graph, the vertex (the peak) is at \((2, 4)\), so \( h = 2 \), \( k = 4 \). So the equation should be \( y = a|x - 2| + 4 \).

Step2: Determine the direction (a's sign)

The graph opens downward (since the left side is increasing and the right side is decreasing), so \( a \) is negative. So \( a = -1 \).

Step3: Form the equation

Substituting \( a = -1 \), \( h = 2 \), \( k = 4 \) into the vertex form, we get \( y = -|x - 2| + 4 \).

Answer:

\( y = -|x - 2| + 4 \) (corresponding to the first option: \( y = -|x - 2| + 4 \))